Impact of conformational structures on primary decomposition of cis-1,2-dimethylcyclohexyl isomers: A theoretical study

Combustion and Flame 205 (2019) 193–205

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Impact of conformational structures on primary decomposition of cis-1,2-dimethylcyclohexyl isomers: A theoretical study Huiting Bian a, Lili Ye b, Jing Li a,∗, Jinhua Sun c,∗, Tianshui Liang a, Wei Zhong a, Jun Zhao a a

School of Mechanics and Engineering Science, Zhengzhou University, Zhengzhou, Henan 450001, PR China School of Science, Dalian Maritime University, Dalian, Liaoning 116026, PR China c State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, Anhui 230026, PR China b

a r t i c l e

i n f o

Article history: Received 26 January 2019 Revised 6 March 2019 Accepted 9 April 2019

Keywords: cis-1,2-Dimethylcyclohexane Conformational structures Decomposition H-transfer Rate coefficients

a b s t r a c t The different orientations of two methyl groups in “strain-free” cyclic structure generate multiple conformational structures for dimethyl-substituted cyclohexanes. These conformational structures are most likely to affect the radical stabilities, activation energies, and rate coefficients of key types of reactions in dimethyl-substituted cyclohexane combustion. The conformational inversion-topomerization mechanism among various conformers for cis-1,2-dimethylcyclohexyl isomers has been explored by applying high-level quantum electronic-structure methods and transition state theory (TST). Intramolecular H-transfers and β -scissions were also investigated to fundamentally unravel the way how the conformational structures impact their initial decomposition. The present kinetic predictions show that conformational changes are much more rapid compared with the primary decomposition of cis-1,2dimethylcyclohexyl isomers. It contributes to the establishment of quasi-equilibrium condition for various conformers retained in each radical and ensures the coexistence of all conformers over 30 0–250 0 K. For the primary decomposition, the intramolecular H-transfers are greatly influenced by the conformational structures. Of particular interest is to observe that 1,4 and 1,5 H-transfers that shift the radical site between side chain and ring are only feasible for chair and twist-boat conformers with the radical site locating in axial side chain. Additionally, the β -scissions of cis-1,2-dimethylcyclohexyl isomers also exhibit the dependence on the conformational structures in aspect of steric energy and substituent effect. Furthermore, facilitated by the speedy equilibration among distinct conformers for each isomer, the contribution of each conformer to kinetic predictions for the initial decomposition was systemically evaluated in terms of the temperature-dependent population for diverse conformers obtained by Boltzmann distribution, and then the appropriate rate parameters for each decomposition type were finally recommended. © 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction Cycloalkanes are one major class of hydrocarbons in commercial fuels, making up 20–30% of jet and diesel fuels and 10–30% of automotive and aviation gasoline [1–6]. The development of alternative energy sources, such as oil-sands and coal-based synthetic fuels, is further increasing the application of cycloalkanes in transportation fuels [5,7,8]. Due to their large content in real fuels, there are many studies focusing on the combustion properties and detailed reaction mechanisms by performing theoretical calculations or experimental measurements [4–6,9–12]. Recently, ∗

Corresponding authors. E-mail addresses: [email protected] (H. Bian), [email protected] (J. Li), [email protected] (J. Sun).

the low temperature oxidation chemistry of cycloalkanes has received increasing attention because of their application in the next generation engine designs [6–8,13,14]. The previous investigations have mainly focused on the study of monoalkylated cycloalkanes with various side chain lengths. Cycloalkanes with multiple alkyl side chains have been barely noticed, and relevant studies remain rather lacking. As suggested by Eldeeb et al. [3], the combustion chemistry of multiple alkylsubstituted cycloalkanes should be investigated in the global perspective of revealing the difference in isomer reactivity and the kinetic effect of alkylation by comparing with monoalkylated cyclohexanes. Recently, McEnally and Pfefferle [15] studied the unimolecular dissociation of mono- and bi-substituted cyclohexanes in coflowing methane/air nonpremixed flame. They found that the decomposition rates of dimethylcyclohexanes are smaller than

https://doi.org/10.1016/j.combustflame.2019.04.024 0010-2180/© 2019 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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Fig. 1. Schematic structures for four cis-1,2-dimethylcyclohexyl radicals: (a) (1methyl-cyclohex-2-yl)-methyl (W1), (b) 1,2-dimethyl-cyclohex-1-yl (W2), (c) 1,2dimethyl-cyclohex-3-yl (W3), and (d) 1,2-dimethyl-cyclohex-4-yl (W4).

those of mono-substituted cyclohexanes. Sun et al. [16] investigated the pyrolysis mechanism of mono-, bi-, and tri-substituted cyclohexanes to reveal the substituent effect by theoretical simulation and experiment. Results showed that the closer two methyl groups are in DMCHs, the more effective it is to increase the formation of gaseous products. Kang et al. [7] revealed the impact of branched structures on ignition characteristics for ethylcyclohexane (ECH) and its two isomers, 1,2-dimethylcyclohexane (1,2DMCH) and 1,3-dimethylcyclohexane (1,3DMCH), in a motored engine. Supported by the conformational analysis, their results revealed that both 1,2DMCH and 1,3DMCH have lower reactivity than ECH. This conclusion was verified by Eldeeb et al. in their study on the ignition and pyrolysis of 1,3DMCH using experiment and kinetic modeling [3]. In addition, several more reports are also available for the reaction kinetics of multiple alkyl-substituted cycloalkanes, including the ring opening reactions of 1,2DMCH and 1,3DMCH by Dokjampa et al. [17,18] and Do et al. [19], the isomerization of cis-1,2-DMCH by Rosado-Reyes and Tsang [20], and the H abstraction reactions of dimethylcyclohexanes and trimethylcyclohexanes by Sway [21]. Although the previous limited studies of dimethylcyclohexanes have touched on the effect of the alkyl substituents on molecular reactivity, they barely encompass a complete consideration of the conformational structures caused by the flexible “strain-free” cyclic

structure, nor the conformational changing reactions among conformers. Moreover, the prior studies remain too weak in uncovering the profound influence of conformational structures for cyclic alkanes on their elementary reactions and consequently on combustion reactivity and behavior. Davis et al. [2] investigated four distinguishable conformers of methyl- and ethylcyclohexyl radicals, and searched out all possible H-transfer channels to obtain their respective kinetic and thermodynamic parameters, aiming for a quantitative determination for the effect of cyclic ring. Yu et al. [22] employed the multiple-path variational transition state theory (MP-VTST) to calculate the forward and reverse thermal rate constants of 1,4 H-transfer for 2-cyclohexyl-ethyl radical. For this new formulation, 22, 9, and 4 distinct conformers are identified for 2-cyclohexyl-ethyl, 2-ethylcyclohexyl, and transition state, respectively, that are included in the multiple paths incorporated to get the accurate kinetic data. According to our previous work, the mechanism of conformational changes greatly influences the distribution of ethylcyclohexyl radicals and thus affects the formation of subsequent decomposition products [9]. Therefore, such conformational structures, comprising of two side chains and the “strainfree” cyclic structure, together with the conformational changes cannot be ignored in the study of dimethylcyclohexane combustion chemistry. The present work aims to explore the impact of conformational structures on the detailed chemistry for several key types of reactions in dimethylcyclohexane combustion. With this motivation, four cis-1,2-dimethylcyclohexyl radicals were chosen as the representatives. The conformational inversion-topomerization reactions among all distinct conformers of four cis-1,2-DMCH radicals were investigated by high-level quantum chemical calculations. Besides, all the H-transfers and β -scissions available for each conformer were also looked into. Kinetic evaluations for these three types of reactions were obtained with transition state theory. In particular, for channels without well-defined saddle points, the thermal rate constants were explored by a variational implementation of transition state theory. In addition, according to the deviations

Fig. 2. Inversion-topomerization process of (1-methyl-cyclohex-2-yl)-methyl (W1). Energies (kcal/mol) of various conformers and transition states are relative to W1CeR a at CCSD(T)/6-311++G(d,p)//B3LYP/6-311++G(d,p) at 0 K (zero-point vibrational energy included).

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Fig. 3. Inversion-topomerization processes of W2-W4. Energies (kcal/mol) of various conformers and transition states are relative to CeR a at CCSD(T)/6311++G(d,p)//B3LYP/6-311++G(d,p) at 0 K (zero-point vibrational energy included).

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in energetics for various conformers of each cis-1,2-DMCH radical, their conformational populations were calculated by using Boltzmann distribution. With this information, the proper rate parameters for each decomposition type studied were suggested after considering the contribution of each conformer in perspective of the thermodynamic stability. 2. Theoretical methodology The equilibrium geometries and vibrational frequencies of all stationary points were achieved at B3LYP/6-311++G(d,p) level, followed by vibrational analysis to confirm the transition states that feature an imaginary frequency. IRC calculations were performed to verify that the transition states connect the corresponding reactants and products properly. Afterwards, the CCSD(T)/6311++G(d,p) method was used to compute high-level single point energies at B3LYP/6-311++G(d,p) geometries. The zero-point vibrational energies were then included from the B3LYP/6-311++G(d,p) frequency analysis. Moreover, previous studies suggested that the composite method G4 [23] is a useful tool to determine energies of unimolecular reactions for cycloalkyl radicals [2,6,9], and thus we also used this method to obtain the energies of all stationary points for comparison purpose. The G4 procedure includes a sequence of single point energy calculations and two new high-level corrections based on equilibrium structure at B3LYP/6-31G(2d,p). For barrierless C–H β -dissociations of various conformers for 1,2-dimethyl-cyclohex-1-yl radical, the zeropoint-inclusive potential energy surface of minimum energy path (MEP) was calculated by B3LYP/6-311++G(d,p) method along the ˚ and the interacdissociation bond with a step size of 0.1 A, tion energies were refined by CCSD(T)/6-311++G(d,p). All quantum chemical computations were performed with the Gaussian 09 program [24]. The high pressure limit rate coefficients for conformational changes, intramolecular H-transfers, and β -scissions for four cis1,2-DMCH radicals were obtained by using transition state theory. During the kinetic calculation, the reaction path degeneracy was included to take into account the chirality numbers of all conformers for reactants, transition states, and products. Particularly for barrierless channels, the variational transition state theory was employed. Since the quantum tunneling effect potentially becomes important at low temperatures, particularly for H-transfer reactions. One-dimensional asymmetric Eckart tunneling [25,26] was incorporated to correct the rate coefficients. Additionally, a fairly large number of torsional modes were identified for the stable species and transition states, the anharmonic effect induced by the internal torsions could be remarkable, especially at high combustion temperatures [27,28]. For these modes, the harmonic oscillator approximation is simply used that introduces large uncertainties. In present work, the low frequency modes corresponding to the internal rotations were treated as one-dimensional hindered rotors. Their hindrance potentials were obtained at B3LYP/6-311++G(d,p) level by applying a relaxed scan along designated dihedral coordinate with an interval of 10°. The computed potentials were then fitted to Fourier series expansions. Moreover, the modes relevant to the pseudorotation of cyclic ring present a challenge to deal with, since they are not the true vibrations [29]. In previous study by Yu et al. [22], they proposed a method to handle these pseudorotation modes. It is to search out all distinct ring structures and then use the harmonic oscillator approximation to treat the vibrational degrees of freedom for each structure. In current work, therefore, all various conformers for cis-1,2-DMCH isomers and transition states were explored, of which the low frequency modes in ring were treated as rigid rotor harmonic oscillators (RRHO). The temperature-dependent rate constants were described with the modified Arrhenius form to

Fig. 4. H-transfer and β -scission activation energies of (1-methyl-cyclohex-2-yl)methyl (W1) conformers relative to W1CeR a at 0 K, computed at CCSD(T)/6311++G(d,p)//B3LYP/6-311++G(d,p) (zero-point vibrational energy included).

facilitate their application in kinetic modeling. Kinetic predictions were obtained by using MESS program [30]. 3. Results and discussion The schematic structures of four cis-1,2-DMCH radicals, i.e. (1-methyl-cyclohex-2-yl)-methyl (W1), 1,2-dimethyl-cyclohex-1-yl (W2), 1,2-dimethyl-cyclohex-3-yl (W3), and 1,2-dimethyl-cyclohex4-yl (W4), are shown in Fig. 1. Their conformational structures are generated by different orientations of two ortho side chains in “strain-free” cyclic structures. With five types of H atoms in cyclohexyl ring mentioned in our prior studies [9,31], the stable chair and twist-boat conformers are labeled by a combination of uppercase (C or T to denote “chair” or “twist-boat”) and lowercase (e, a, or i to denote “equatorial”, “axial” or “isoclinal”). “R” is further added as superscript to indicate the location of radical site for W1, or for W2–W4 to denote the side chain that is closer to the radical site. For example, W1CeR a represents the chair conformer in W1 with the radical site at equatorial side chain and the methyl in axial position. For W2-W4, whereas, CeR a represents the chair conformer in which the radical site is closer to the equatorial methyl than the axial one. Half-chair and boat transition state

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Fig. 5. H-transfer and β -scission activation energies of 1,2-dimethyl-cyclohex-1-yl (W2) and 1,2-dimethyl-cyclohex-3-yl (W3) conformers relative to W1CeR a at 0 K, computed at CCSD(T)/6-311++G(d,p)//B3LYP/6-311++G(d,p) (zero-point vibrational energy included).

structures are labeled by a combination of uppercase (H and B) and number, for example, H1. H-transfer is identified by a usual format Ncd, where N is the ring size formed in transition states, c and d represent the types of radical sites for product and reactant (p = primary, s = secondary, and t = tertiary), respectively. C–C or C–H β -scission is denoted by β CC or β CH, respectively, as shown in Figs. S9-S12 of the Supplemental material-I. For each cis-1,2DMCH radical, the optimized structures of distinct conformers and their conformational transition states are individually illustrated in Figs. S1–S8 of the Supplemental material-I. The Cartesian coordinates and frequencies of the optimized structures for all reactants, transition states, and products are provided in the Supplemental material-II. And thus their energies are listed in Tables S1–S3 of the Supplemental material-I. The deviations in energy between the two methods are less than 0.5 and 2 kcal/mol for conformers and transition states, respectively. The CCSD(T)/6-311++G(d,p) values are utilized in rate constant calculations and also in the following discussion considering its good performance in hydrocarbon combustion chemistry [32,33]. Additionally, the thermochemical data of all conformers for four cis-1,2-DMCH radicals and products were calculated by the group additivity method carried out with Thergas

software. These results were then fitted into the NASA polynomial form provided in the Supplemental material-III. 3.1. Conformational analysis The presences of two side chains in the cyclohexane ring and the varying location of radical site lead to distinct conformational behaviors for cis-1,2-DMCH radicals. Briefly, the inversion-topomerization process for W1 is firstly discussed. As shown in Fig. 2, the inversion process of conformers follows the sequence CeR a-H1-TeR i-B9-TeiR -H2-CeaR -H3-TeaR . It can be seen that the interconversion of two chair conformers (i.e. CeR a and CeaR ) is achieved by going through twist-boat conformers TeR i and TeiR . Note that there is no transition state serving to connect CeR a to TeR a. According to our prior work [31], it might have high steric repulsion that cannot stable exist and be captured by the current DFT calculation. The topomerization process of twist-boat conformers is independent of the inversion process [31,34], following the sequence TeR i-B4-TeR a-B5-TiR a-B6-TiaR -B7-TeaR -B8-TeiR -B9TeR i. The half-chair transition states H1-H3 connecting chairs and twist-boats have higher barrier heights (10.1–12.3 kcal/mol)

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Fig. 6. The β -scission activation energies of 1,2-dimethyl-cyclohex-4-yl (W4) conformers relative to W1CeR a at 0 K, computed at CCSD(T)/6-311++G(d,p)//B3LYP/6311++G(d,p) (zero-point vibrational energy included).

than boats B4-B9 (6.8–10.1 kcal/mol) connecting twist-boats. It implies that the inversion process is the controlling step of the whole conformational process. This conformational trend is in accordance with those of methylcyclohexane and 2-cyclohexyl-

Fig. 7. Temperature dependent rate constants of conformational changes, Htransfers, and β -scissions for W1CeaR .

ethyl [9,34]. Moreover, of particular interest is the observation that this trend gives an excellent agreement with that of cis-1,2dimethylcyclohexane predicted in our prior work [31]. Note that one H atom loss in side chain gives rise to more distinct conformers. Moreover, compared with monocyclohexyl radicals, W1 shows

Fig. 8. Boltzmann distribution of four 1,2-dimethylcyclohexyl radicals (W1–W4) conformers over 30 0-250 0 K.

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a different conformational behavior due to the substituent effect. It is obviously noticed that H3 for W1 is 10.1 kcal/mol, while its corresponding structure H7 for 2-cyclohexyl-ethyl is 12.7 kcal/mol [9] because of its higher steric energy raised by the repulsion between the long ethyl group and H atoms on ring. There are two chairs (−0.6 and 0 kcal/mol) and six twist-boats (5.4–8.2 kcal/mol) for W1, of which the structures are shown in Fig. S1 of the Supplemental material-I. 2-Cyclohexyl-ethyl, as one isomer of W1, has more conformers, i.e. four chairs (0–1.8 kcal/mol) and eight twist-boats (6.1–7.7 kcal/mol) [9]. It is caused by the fact that the internal rotation of the side ethyl group can produce extra more conformers for the chair and twist-boat ring structures. This phenomenon reminds us that the rotation of long side chain should be an important factor that affects the molecular structures for alkyl-substituted cycloalkanes [31]. Among the eight conformers of W1, the chair conformer CeaR with the radical site locating in the axial side chain has the lowest energy (i.e. −0.6 kcal/mol), which is slightly lower than that of CeR a, as shown in Fig. 2. One H atom loss in the axial side chain reduces more repulsion between the side chain and H atoms on ring. Similarly, the twist-boat conformer TiaR (6.8 kcal/mol) with the radical site locating in axial position releases more steric energy by comparison with TiR a (8.2 kcal/mol). In addition, the inversion-topomerization processes of W2–W4 radicals were also explored in the same way, of which the twodimensional schemes of energetics are demonstrated in Fig. 3. It is obvious that the zero-point-inclusive potential energy surfaces of W2–W4 (with radical site on the ring) are less complicated by comparison with that for W1 (with radical site in side chain). Only six, five, and five stable conformers are captured for W2, W3, and W4, respectively, less than the conformer number (i.e. eight) of W1. The same situation also happens to transition states that connect these stable conformers. This similar trend is observed in the prior studies [9,22] that 2-cyclohexyl-ethyl (with radical site locating in side chain) and 2-ethylcyclohexyl (with radical site locating in cyclic ring) have 22 and 9 conformers, respectively, as mentioned above. In addition, the energy deviations between the conformers and their conformational transition states for W2–W4 are also smaller than those for W1, as is deduced from Table S1 of the Supplemental material-I. The reason for these phenomena is that W2–W4 with one H atom less on cyclic ring have a weakened repulsion among H atoms on ring, resulting in the cyclic structure becoming more easily distorted. Moreover, it should be pointed out that the methyl group and radical site in the same location on ring would release more strain. Seen from Fig. 3, these transition states involved in conformational process of W2 have the lower energies than those for W3 and W4. Therefore, the location of radical site could influence the stability of W1–W4. Furthermore, zero-pointinclusive potential energy surfaces containing distinct conformers for W1–W4 are used to compute the rate coefficients of conformational changing reactions. It aims to further address the impact of conformational structures from a kinetic perspective.

3.2. H-transfer and β -scission The primary decomposition reactions of cis-1,2-DMCH radicals studied in present work, mainly including H-transfer and β scission, are discussed separately in the following sections. According to previous studies, H-transfer plays a significant role in hydrocarbon combustion since it can relocate the radical site and affect the formation of the decomposition products [2,9]. Recently, Kang et al. [7] and Yang et al. [8,13] reported that 1,4 and 1,5 H-transfers are the key steps in low temperature oxidation of cyclic hydrocarbons by terminating or degenerating chain branching. Since there is a common view that H-transfer highly relies on the molecular

Fig. 9. Temperature dependent rate constants of H-transfers (a) and β -scissions (b) for (1-methyl-cyclohex-2-yl)-methyl (W1) conformers.

structure [2,7–9,13,22], various types of H-transfers available for all distinct conformers of W1–W4 radicals were studied theoretically. Figure 4 depicts the activation energies of various types of H-transfers and β -scissions for the eight conformers of W1. In previous work by Davis et al. [2], a ring-based effect is observed that the ring structures in chair and boat configurations for methyl- and ethylcyclohexyl radicals significantly influences on the activation energies of H-transfers. Their results showed that,

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Fig. 10. Temperature dependent rate constants of H-transfers (a) and β -scissions (b–d) for 1,2-dimethyl-cyclohex-1-yl (W2) conformers.

for the same type of H-transfers, the deviations in the activation energies are similar to the energy disparities between their respective boat and chair conformers. Note that the energies of stable chair configurations for both radicals, having the primary radicals locating in equatorial side chains, are utilized as the references for these activation energies. This is very conducive to directly demonstrate such ring-based effect for diverse conformers in figures. Therefore, in current work the energies of transition states for the initial decomposition are referred to the chair form with the primary radical (W1CeR a). From Fig. 4(a), all conformers can undergo 1,2 H-transfers, among which the barriers for twist-boat conformers (42.8–45.2 kcal/mol) are higher than those for chair ones (38.4–39.3 kcal/mol). This is in accordance with the energy trend of W1 conformers. For 1,3 H-transfers, there is an interesting phenomenon that W1CeaR , W1TeaR , and W1TiR a have much lower activation energies than the other three conformers. It results from the fact that the conformers with radical site locating in axial or isoclinal position can facilitate the 1,3 H-transfers. Note that there is no 1,3 H-transfer available for W1TeiR and W1TiaR . Additionally, the important 1,4 and 1,5 H-transfers shifting radical site between side chain and cyclic ring can only happen when the radical site is initially located at the axial side chain, i.e., 1,4 H-transfers for the W1CeaR , W1TeaR , and W1TiaR conformers and 1,5 H-transfer for W1TeaR only. Moreover, there is one type of 1,4 H-transfers occurring between the two side chains, i.e., isomerization between W1CeR a and W1CeaR , W1TeR i and W1TeiR , and W1TiR a and W1TiaR . Figure 4(a) only displays the activation en-

ergies for one conformer in each pair, i.e., W1CeR a, W1TeR i, and W1TiR a. These results drive us to conclude that H-transfers for W1 are strongly influenced by the location of radical site and ring structure, and the conformers with the radical site in axial side chain play a more important role. Moreover, it is apparently noticed that 8, 7, 7, and 1 transition states are identified for 1,2 to 1,5 H-transfers, respectively. For β -scissions, all stable conformers can undergo three decomposition channels shown in Fig. 4(b). But indeed, there are 6, 5, and 6 distinct transition state structures involved in β CC1, β CC2, and β CH3 reactions. For each type of β -scissions, the activation energies vary markedly with the energies of the corresponding conformers. This is because the steric energies for various conformers remain in their relevant transition states and cannot be completely released by breaking chemical bonds. Moreover, for each conformer, the cleavage of C-C bond between two side chains labeled as β CC2 has the lowest barrier height. According to the Evans-Polanyi correlations [35,36], this phenomenon may be caused by the fact that the secondary radical generated from W1β CC2 is more stable than the primary radical for W1β CC1 and the separate products for W1β CH3, as listed in Table S3 of the Supplemental material-I. For W2-W3, the H-transfers show slight variation in energies of the transition states corresponding to various conformers. However, the barriers for 1,4 H-transfers generating from W2 are obviously lower than these for 1,2 and 1,3 H-transfers, as

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Fig. 11. Temperature dependent rate constants of H-transfers (a) and β -scissions (b–d) for 1,2-dimethyl-cyclohex-3-yl (W3) conformers.

depicted in Fig. 5(a). Note that 1,4 H-transfers are merely feasible for W2TeR i, W2TeaR , and W2TiaR . It implies that such reaction paths are also greatly influenced by the conformational structures, and their activation energies are listed in Table S2 of the Supplemental material-I. In addition, the similar conformational effect on activation energies of β -scissions for W2-W4 is also observed in Figs. 5 and 6. Their detailed reaction schemes are demonstrated in Figs. S10-S12 of the Supplemental material-I. The deviations in energies of these transition states for each type of dissociations are larger than those for the former H-transfers. It should be mentioned that one H atom loss in methyl group proceeds by C–H β -scission that is the barrierless reaction without obvious saddle point, as presented in Fig. S10 of the Supplemental materialI. Among different β -scission types for W2–W4, the reactions marked by W2β CC3, W3β CC1, and W4β CC2 have the lowest energy barriers. Same to W1β CC2, W3β CC1 also yields one secondary radical that has the most stable structure by comparison with the other alkenyl radicals in Table S3 of the Supplemental material-I, making this reaction channel more favorable. The quantum chemical results for the primary decomposition of W1–W4 were used in kinetic calculations in the following section. 3.3. Kinetic calculations Based on the quantum chemical calculations above, temperature-dependent rate coefficients of conformational changes, H-transfers, and β -scissions, which are relative to

each conformer of W1-W4, were computed by transition state theory to further explore the effect of conformational structures on combustion kinetics. The kinetic data in the modified Arrhenius form are listed in Tables S4 and S5 of the Supplemental material-I. Figure 7 plots the temperature-dependent rate coefficients of the conformational changes, H-transfers, and β -scissions for W1CeaR over 40 0–250 0 K. Since there is no previous report on cis-1,2dimethylcyclohexyl radicals, the kinetic results of H-transfers [2] and β -scissions [37] for methylcyclohexyl are used for comparison considering the great similarity in their structures. For β -scissions, the rate constants of methylcyclohexyl are one to two times larger than those of W1CeaR because of the energy disparity in barriers between them (within 1 kcal/mol). The rate coefficients of 1,4 H-transfers are in good consistency with the data of Davis et al., except those below 500 K. It might result from the fact that 1,4 H-transfers studied in current work has slight higher barriers than that from Davis et al. [2]. Moreover, the quantum tunneling effect was considered by using one-dimensional asymmetric Eckart’s potential in this study, while those reported by Davis et al. were corrected by the methods proposed by Skodje et al. [38], and by Wigner [39]. For 1,2 H-transfers, our kinetic results are two to nine times smaller than their relevant data below 10 0 0 K, since our barrier is roughly 2 kcal/mol higher. For the primary decomposition of W1CeaR , the three 1,4 Htransfers are more favored than the others below 800 K, which is attributed to the lower barrier heights (25.3–25.6 kcal/mol). Above 800 K, two C–C β -scissions become competitive with 1,4 H-

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Table 1 High-pressure limit rate parameters suggested for each type of H-transfers and β -scissions for four cis-1,2-dimethylcyclohexyl radicals in the modified Arrhenius equation k=ATn exp(−Ea /RT). Species

Reactions

(1-methyl-cyclohex-2-yl)-methyl (W1)

H-transfers W1→W2 (W1CeR a3tp) W1→W2 (W1CeR a4tp) W1→W3 (W1CeaR 4sp) W1→W3 (W1CeaR 5sp) W1→W4 (W1CeaR 5sp) W1→W4 (W1TeaR 6sp) Dissociations W1CeaR β CC1 W1CeaR β CC2 W1CeaR β CH3 H-transfers W2→W3 (W2TeR i3st) W2→W3 (W2TeaR 4st) W2→W4 (W2CeaR 4st) W2→W4 (W2TeaR 5st) Dissociations W2CeaR β CC1 W2CeaR β CC2 W2CeR aβ CC3 W2CeR aβ CH4 W2TiR aβ CH5 W2CeaR β CH6 H-transfers W3→W4 (W3CeaR 3ss) W3→W4 (W3CeaR 4ss) Dissociations W3CeR aβ CC1 W3CeR aβ CC2 W3CeaR β CC3 W3TeaR β CH4 W3CeaR β CH5 Dissociations W4CeaR β CC1 W4CeaR β CC2 W4CeaR β CH3 W4CeaR β CH4

1,2-dimethyl-cyclohex-1-yl (W2)

1,2-dimethyl-cyclohex-3-yl (W3)

1,2-dimethyl-cyclohex-4-yl (W4)

transfers and play a leading role because of their large entropies. However, neither H-transfers nor β -scissions can compete with the conformational changing reactions. As depicted in Fig. 7, two conformational changes of W1CeaR directly connecting to W1TeiR and W1TeaR are several orders of magnitude faster than its Htransfers and β -scissions over 40 0–250 0 K. This can be ascribed to their much lower barriers (i.e. 10.1 and 12.3 kcal/mol). Generally, the rate coefficients of conformational inversion-topomerization reactions for each conformer of W1–W4 radicals are several orders of magnitude larger than those for the decomposition over the studied temperature range, which is in agreement with our previous results for ethylcyclohexyl radicals [9]. Therefore, the same conclusion can be derived in present work that the conformational changes are not the rate controlling steps compared with initial decomposition of cis-1,2-DMCH radicals. The fairly fast transformation among distinct conformers for each radical is most likely to establish quasi-equilibrium condition before they undergo subsequent consumption pathways. It proves that all conformers can co-exist in combustion. Moreover, the inversion-topomerization has an important contribution to entropies of W1-W4 radicals by considering all unique conformational structures produced by the pseudorotation of ring [22]. Note that the H-transfers serving to connect conformers for one radical, for instance 1,4 H-transfer between W1CeR a and W1CeaR , could be ignored safely in combustion kinetics for W1–W4 by comparing with the rapid conformational changes. As is mentioned, the equilibration among all diverse conformers of each radical is established by the rapid conformational changes prior to the primary decomposition. It conduces to entirely con-

A (s−1 )

n

Ea (kcal/mol)

8.73E+03 1.15E+02 8.31E+02 3.45E+03 3.41E+03 7.70E+03

0.427 0.867 0.683 0.434 0.438 0.343

15.6 19.5 16.1 9.8 9.7 9.6

7.37E+05 8.74E+05 1.08E+05

−0.038 −0.071 0.172

14.2 13.3 16.0

1.16E+04 6.54E+01 1.12E+03 7.41E+03

0.40 1.24 0.67 0.45

16.8 17.3 16.8 13.6

1.04E+06 1.37E+06 1.71E+06 3.36E+05 1.45E+05 4.21E+04

−0.036 −0.057 −0.023 0.137 0.412 0.348

13.9 13.9 13.4 15.8 15.9 17.7

1.14E+04 9.71E+02

0.464 0.685

17.1 17.1

1.57E+06 1.09E+06 1.41E+06 6.98E+03 3.00E+05

−0.082 −0.030 −0.009 0.755 0.148

12.8 14.2 14.0 15.7 15.4

1.32E+06 1.67E+06 3.76E+05 1.81E+04

−0.049 −0.078 0.140 0.735

14.5 13.6 15.3 15.6

sider the contribution of each conformer to the initial dissociations in terms of their thermodynamic stability. On base of this, the multiple paths involved in one decomposition type could be comprehensively compared to find the most favorable channel. Therefore, with the energies for W1–W4 radicals depicted in Table S1 of the Supplemental material-I, the temperature-dependent populations for all conformers of each radical were calculated by the Boltzmann distribution over 30 0–250 0 K. These values will then multiply the corresponding rate constants for the same conformer to obtain the final kinetic data that can be regarded as being calculated from the most stable conformer in a whole new perspective. In previous work by Yu et al. [22], the weighting factor, i.e. the rotational-vibrational partition function for distinct transition states, is utilized to evaluate the contribution of paths with different barriers to the path-averaged generalized transmission coefficient in MP-VTST to get the accurate rate parameters. As shown in Fig. 8, the two chair conformers for four cis-1,2-DMCH radicals are the most abundant components over the studied temperature range. However, their populations would decrease with increasing temperature, while the opposite trend is observed for the other conformers simultaneously. For example, the sum of their populations for W1 is 99.5% at 500 K, and then decreases to 89.4% at 10 0 0 K and 63% at 20 0 0 K. The decreasing trend with temperatures is very similar to that of the three chair conformers of 2cyclohexyl-ethyl and ethylcyclohexane reported in our prior studies [9,31]. Furthermore, twist-boat conformers for W1–W4 are scarce below 10 0 0 K, seemingly indicating that they could be ignored under that condition, but these ones have a significant effect on the kinetic predictions.

H. Bian, L. Ye and J. Li et al. / Combustion and Flame 205 (2019) 193–205

As discussed above, the activation energies of various Htransfers and β -scissions for cis-1,2-DMCH radicals are greatly influenced by the conformational structures. Herein, their impact on kinetics is discussed briefly. The kinetic results for W1 with considering the conformers’ contribution are displayed in Fig. 9. It can be seen that the disparities in rate coefficients of 1,2, 1,3, and 1,4 H-transfers are one to four orders of magnitude over 40 0–250 0 K. With respect to 1,3 H-transfers, the conformers with radical sites at equatorial side chains (i.e., W1CeR a, W1TeR i, and W1TeR a) are disadvantageous resulting from the higher barriers, as shown in Fig. 4(a). For 1,4 H-transfers, the largest deviation in rate coefficients occurs at the low temperature end where the activation energy dominates. Over the temperatures of interest here, two 1,4 H-transfer channels for W1CeaR are more favorable than the others, and the 1,5 H-transfer for W1TeaR has the most largest rate coefficients. Note that all transition states for 1,2 to 1,5 H-transfers are formed by locking only one free rotor into the new generated cyclic ring. This means that there is no big difference in entropies for these channels. So, compared with entropies, the disparities in activation energies of 1,2 to 1,5 H-transfers primarily dominate the deviations in their rate constants, particularly at low temperatures. Generally, the ranking of their relative importance for W1 is: 1,5 > 1,4 > 1,2 > 1,3 H-transfers. It is in line with the results for methyl- and ethylcyclohexyl radicals reported by Davis et al. [2]. For β -scissions, the rate coefficients are also apparently affected by the conformational structures, in which the deviation is two orders of magnitude, as shown in Fig. 9(b) and (c). Among these dissociation channels, the rate coefficients of β CC2 are the largest for each conformer. It implies that the cleavage of C-C bond between the two substituents is the most facilitated pathway. This conclusion agrees well with the previous results on 1,2DMCH and 1,3DMCH [7,19]. Do et al. [19] proposed that the presence of methyl substituents more favors the scission of C-C bond between two side chains than the other C-C bonds in ring. Additionally, the radical site locating in axial side chain for W1CeaR always leads to β -scissions with the largest rate constants, as demonstrated in Fig. 9(b) and (c). In addition, C-C β -scissions favor over C-H β -scissions for the studied temperature range, which is in accordance with the conclusion for alkenyl radicals derived from our prior work [40]. For 1,2-dimethyl-cyclohex-1-yl (W2), the decomposition scheme is depicted in Fig. S10 of the Supplemental material-I and the kinetic results are displayed in Fig. 10. From Fig. 10(a), it is obviously noticed that three 1,4 H-transfer channels separately generated from W2TeR i, W2TeaR , and W2TiaR have the similar rate constants, and they are more favorable than 1,2 and 1,3 H-transfers for the temperature range of 40 0–250 0 K. Note that the rate coefficients of 1,2 and 1,3 H-transfers increase more sharply than those for 1,4 Htransfers due to their larger entropies. For bond dissociations, C-C and C-H β -scissions each have three different types, of which the rate coefficients are respectively depicted in Fig. 10(b)–(d) for clarity purpose. As mentioned, there is one barrierless reaction type marked by β CH6 in Fig. 10(c). For all the dissociations, six pathways labeled as W2β CC3 have larger rate constants than the others, which are contributed by the lower barriers shown in Fig. 5(b). Moreover, one W2β CC3 channel for W2CeR a conformer has the largest rate coefficients over 40 0–250 0 K. It means that this channel is the main consumption pathway for W2. For the other types of C-C β -scissions, two paths indicated by W2β CC1 and W2β CC2 generated from W2CeaR are most facilitated. In addition, the conformational structures W2CeR a, W2TiaR , and W2CeaR are in favor of β CH1, β CH2, and β CH3, respectively, as listed in Table 1. Additionally, W3 and W4 have seven and four initial decomposition pathways, of which the reaction schemes are displayed in Figs. S11 and S12 of the Supplemental material-I, respectively. With regard to the kinetic results for W3 in Fig. 11, we can see that 1,2 and 1,3 H-transfers are less competitive by comparison with bond

203

Fig. 12. Temperature dependent rate constants of H-transfers (a) and β -scissions (b) for 1,2-dimethyl-cyclohex-4-yl (W4) conformers.

dissociations over 40 0–250 0 K. It is ascribed to the combined effect of the higher barriers and smaller entropies. For β -C-C scissions, W3β CC1, W3β CC2, and W3β CC3 are favored by the conformers W3CeR a, W3CeR a, and W3CeaR accordingly. Same to W1β CC2, W3β CC1 for W3CeR a has the largest rate coefficients with the ring opening of substituted C-C bond to produce a secondary radical mentioned above. It potentially plays a significant role in the overall decomposition mechanism of W3. Furthermore, the most stable conformer W3CeR a also conduces to undergo two classes of C-H β -scissions, as shown in Fig. 11(b) and (d). For W4, all the decomposition pathways are facilitated by the conformational structure W4CeaR , especially the one corresponding to W4β CC2 having the largest rate constants over 40 0–250 0 K, as displayed in Fig. 12. According to all the discussions above, it can be obviously found that the most favored decomposition channels for W1-W4 radicals are yielded from the conformers W1CeaR , W2CeR a, W3CeR a, and W4CeaR , respectively. Note that these four conformers correspondingly have the greatest populations over 30 0–250 0 K depicted in Fig. 8. It could be inferred that these more stable structures have the major impact on these predicted rates. Moreover, it is worth mentioning that, the decomposition of one radical will affect the consumption of the other radicals through inclusion of the important H-transfers facilitated by the distinct conformers. For instance, 1,5 H-transfer of W1TeaR , playing an important role in con-

204

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sumption of W1 below 900 K, connects to W4TeR a and can thus influence the subsequent decomposition of W4 via the inversiontopomerization mechanism. According to our previous work [9], it could better demonstrate the effect of the conformational structure on kinetic behaviors, such as radical concentration and product formation, by incorporating the conformational changes, H-transfers, and β -scissions of W1–W4 into a kinetic model. Therefore, to facilitate the application of the present results in combustion model, the most appropriate rate parameters for each decomposition type investigated are summarized in Table 1. These kinetic predictions are certainly affected by distinct conformational structures for four cis-1,2-DMCH isomers, and would be utilized to conduct the kinetic simulation to compare with the available experimental results in future study. 4. Conclusions The conformational inversion-topomerization reactions among distinct conformers for four cis-1,2-dimethylcyclohexyl isomers were studied by quantum chemical calculations and kinetic predictions. Additionally, the initial decomposition for each conformer, including intramolecular H-transfers and β -scissions, were also looked into by the same methodologies. Based on these theoretical studies, the impact of conformational structures on radical stabilities, activation energies, and rate coefficients for the primary decomposition of cis-1,2-dimethylcyclohexyl radicals were revealed. The findings in present work indicate that the conformational changes, which are much more rapid than intramolecular H-transfers and β -scissions, end up in establishing a quasi-equilibrium condition for distinct conformers with the same radical site. This condition allows comparing multiple paths yielded from various conformers for one decomposition type by thoroughly considering the contribution of each conformer. Moreover, the conformational structures have a significant entropy contribution to the pseudorotation of ring structure in perspective of thermodynamics. Compared to conformational changes, H-transfers connecting conformers for one radical could be ignored in combustion kinetics for W1-W4. Of particular interest is observation that 1,2 to 1,5 Htransfers are greatly affected by the conformational structures. For instance, 1,5 H-transfer only occurs in W1TeaR . Note that it could be an important consumption pathway for W1 by transforming to W4TeR a below 900 K. The consumption of W4, in this way, can be affected by W1. Additionally, the β -scissions between two side groups (labeled as β CC2 for W1CeaR in this work) are the most favored decomposition pathways of W1 above 900 K because of the substituent effect, and the same situation happens to W3β CC1 generated from W3CeR a that plays a significant role in the decomposition of W3. W2β CC3 for W2CeR a dominates the consumption of W2 and W4β CC2 of W4CeaR has the largest rate coefficients over 40 0–250 0 K. Furthermore, based on all the available reactions studied for each conformer, the most suitable kinetic data are recommended for each decomposition type with regard to the conformational populations of distinct conformers that are calculated by Boltzmann distribution. These results allow construction of the kinetic model for cis-1,2-dimethylcyclohexyl radicals to further explore the effect of conformational structures on the end decomposition products in future study. And the related experimental data are certainly desired. Acknowledgments Authors are grateful for the funding support from National Natural Science Foundation of China (51606122, 11602226, and 51874258), Open Research Fund of State Key Laboratory of Fire Science (No. HZ2019-KF01), and Program for Changjiang Scholars and

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